## Principal nilpotent orbits and reducible principal series

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- by Wentang Kuo
- Represent. Theory
**6**(2002), 127-159 - DOI: https://doi.org/10.1090/S1088-4165-02-00132-2
- Published electronically: July 25, 2002
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## Abstract:

Let $G$ be a split reductive $p$-adic group. In this paper, we establish an explicit link between principal nilpotent orbits of $G$ and the irreducible constituents of principal series of $G$. A geometric characterization of certain irreducible constituents is also provided.## References

- Dan Barbasch and David A. Vogan Jr.,
*The local structure of characters*, J. Functional Analysis**37**(1980), no. 1, 27–55. MR**576644**, DOI 10.1016/0022-1236(80)90026-9 - A. Borel,
*Automorphic $L$-functions*, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 27–61. MR**546608**, DOI 10.1090/pspum/033.2/546608
B Bourbaki, N., “Groupes et algèbres de Lie,” Chapters 7 and 8. Fasc. XXXVIII, Paris, Hermann 1975.
- F. Bruhat and J. Tits,
*Groupes réductifs sur un corps local*, Inst. Hautes Études Sci. Publ. Math.**41**(1972), 5–251 (French). MR**327923**, DOI 10.1007/BF02715544 - W. Casselman and J. Shalika,
*The unramified principal series of $p$-adic groups. II. The Whittaker function*, Compositio Math.**41**(1980), no. 2, 207–231. MR**581582** - Harish-Chandra,
*Admissible invariant distributions on reductive $p$-adic groups*, Lie theories and their applications (Proc. Ann. Sem. Canad. Math. Congr., Queen’s Univ., Kingston, Ont., 1977) Queen’s Papers in Pure and Appl. Math., No. 48, Queen’s Univ., Kingston, Ont., 1978, pp. 281–347. MR**0579175** - S. S. Gelbart and A. W. Knapp,
*Irreducible constituents of principal series of $\textrm {SL}_{n}(k)$*, Duke Math. J.**48**(1981), no. 2, 313–326. MR**620252**, DOI 10.1215/S0012-7094-81-04818-3 - S. S. Gelbart and A. W. Knapp,
*$L$-indistinguishability and $R$ groups for the special linear group*, Adv. in Math.**43**(1982), no. 2, 101–121. MR**644669**, DOI 10.1016/0001-8708(82)90030-5 - I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro,
*Representation theory and automorphic functions*, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. Translated from the Russian by K. A. Hirsch. MR**0233772** - Charles David Keys,
*On the decomposition of reducible principal series representations of $p$-adic Chevalley groups*, Pacific J. Math.**101**(1982), no. 2, 351–388. MR**675406**, DOI 10.2140/pjm.1982.101.351 - C. David Keys,
*$L$-indistinguishability and $R$-groups for quasisplit groups: unitary groups in even dimension*, Ann. Sci. École Norm. Sup. (4)**20**(1987), no. 1, 31–64. MR**892141**, DOI 10.24033/asens.1523 - C. David Keys and Freydoon Shahidi,
*Artin $L$-functions and normalization of intertwining operators*, Ann. Sci. École Norm. Sup. (4)**21**(1988), no. 1, 67–89. MR**944102**, DOI 10.24033/asens.1551 - A. W. Knapp,
*Commutativity of intertwining operators*, Bull. Amer. Math. Soc.**79**(1973), 1016–1018. MR**333074**, DOI 10.1090/S0002-9904-1973-13308-7 - A. W. Knapp,
*Commutativity of intertwining operators. II*, Bull. Amer. Math. Soc.**82**(1976), no. 2, 271–273. MR**407203**, DOI 10.1090/S0002-9904-1976-14019-0 - A. W. Knapp and E. M. Stein,
*Irreducibility theorems for the principal series*, Conference on Harmonic Analysis (Univ. Maryland, College Park, Md., 1971), Lecture Notes in Math., Vol. 266, Springer, Berlin, 1972, pp. 197–214. MR**0422512**, DOI 10.1007/BFb0059645 - Bertram Kostant,
*On Whittaker vectors and representation theory*, Invent. Math.**48**(1978), no. 2, 101–184. MR**507800**, DOI 10.1007/BF01390249 - Hisayosi Matumoto,
*$C^{-\infty }$-Whittaker vectors corresponding to a principal nilpotent orbit of a real reductive linear Lie group, and wave front sets*, Compositio Math.**82**(1992), no. 2, 189–244. MR**1157939** - C. Mœglin and J.-L. Waldspurger,
*Modèles de Whittaker dégénérés pour des groupes $p$-adiques*, Math. Z.**196**(1987), no. 3, 427–452 (French). MR**913667**, DOI 10.1007/BF01200363 - F. Rodier,
*Modèle de Whittaker et caractères de représentations*, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974), Lecture Notes in Math., Vol. 466, Springer, Berlin, 1975, pp. 151–171 (French). MR**0393355**, DOI 10.1007/BFb0082205 - P. J. Sally Jr. and M. H. Taibleson,
*Special functions on locally compact fields*, Acta Math.**116**(1966), 279–309. MR**206349**, DOI 10.1007/BF02392818 - Freydoon Shahidi,
*Functional equation satisfied by certain $L$-functions*, Compositio Math.**37**(1978), no. 2, 171–207. MR**498494** - Freydoon Shahidi,
*On certain $L$-functions*, Amer. J. Math.**103**(1981), no. 2, 297–355. MR**610479**, DOI 10.2307/2374219 - Freydoon Shahidi,
*A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups*, Ann. of Math. (2)**132**(1990), no. 2, 273–330. MR**1070599**, DOI 10.2307/1971524 - Robert Steinberg,
*Lectures on Chevalley groups*, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR**0466335** - David A. Vogan Jr.,
*Gel′fand-Kirillov dimension for Harish-Chandra modules*, Invent. Math.**48**(1978), no. 1, 75–98. MR**506503**, DOI 10.1007/BF01390063

## Bibliographic Information

**Wentang Kuo**- Affiliation: Department of Mathematics, Purdue University, MATH 726, West Lafayette, Indiana 47906
- Address at time of publication: Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada, K7L 3N6
- MR Author ID: 698451
- Email: wtkuo@mast.queensu.ca
- Received by editor(s): July 15, 2001
- Received by editor(s) in revised form: April 11, 2002
- Published electronically: July 25, 2002
- © Copyright 2002 American Mathematical Society
- Journal: Represent. Theory
**6**(2002), 127-159 - MSC (2000): Primary 22E50; Secondary 22E35
- DOI: https://doi.org/10.1090/S1088-4165-02-00132-2
- MathSciNet review: 1915089